Find the domain of the function and identify any vertical and horizontal asymptotes. $$f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$$
A)Domain: all real numbers x Vertical asymptote: No vertical asymptote
Horizontal asymptote: y = 5
B)Domain: all real numbers except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -8
C)Domain: all real numbers x Vertical asymptote: x = -5
Horizontal asymptote: y = 0
D)Domain: all real numbers except x = 5 Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
E)Domain: all real numbers except x = 9 Vertical asymptote: x = 5
Horizontal asymptote: y = 8

Question

Find the domain of the function and identify any vertical and horizontal asymptotes.​ $$f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$$ ​
A)Domain: all real numbers x Vertical asymptote: No vertical asymptote
Horizontal asymptote: y = 5
B)Domain: all real numbers except x = 0 Vertical asymptote: x = 0
Horizontal asymptote: y = -8
C)Domain: all real numbers x Vertical asymptote: x = -5
Horizontal asymptote: y = 0
D)Domain: all real numbers except x = 5 Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
E)Domain: all real numbers except x = 9 Vertical asymptote: x = 5
Horizontal asymptote: y = 8

Quizplus · Accepted Answer

The domain of the function is all real numbers because the denominator $x^2 + x + 9$ does not equal zero for any real value of x (it's always positive due to the nature of quadratic functions with a positive leading coefficient and a positive constant term). There are no vertical asymptotes for the same reason. The horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator, which is $5/1 = 5$, hence $y = 5$.

Find the Domain of the Function and Identify Any Vertical f(x)=5x2+8x2+x+9f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }f(x)=x2+x+95x2+8​

Find the Domain of the Function and Identify Any Vertical $f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$